From: Albert_Rich@msn.com   
      
   On Friday, September 22, 2017 at 6:58:04 AM UTC-10, clicl...@freenet.de wrote:   
   > Albert Rich schrieb:   
   > >    
   > > As Martin pointed out, when m=0 and n=1 or when m=2 and n=3 these   
   > > integrals are pseudo-elliptic.  However for other even values of m,   
   > > the optimal antiderivatives do seem to require a single, simple   
   > > elliptic term.  Do you concur?   
   >    
   > I suppose this is most easily answered by means of a capable algebraic   
   > Risch integrator.   
      
   For the antiderivative of x^2/((2-x^2)*(x^2-1)^(1/4)) Rubi 4.13.3 gets   
      
     ArcTan[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] +    
     ArcTanh[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] -    
     2*(1-x^2)^(1/4)/(x^2-1)^(1/4)*EllipticE[ArcSin[x]/2,2]   
      
   involving only a single EllipticE function.  If this was actually a   
   pseudo-elliptic integral, that would imply EllipticE[ArcSin[x]/2,2] could be   
   expressed in terms of elementary functions.  But surely that is not the   
   case(?).  Ergo the integral is    
   elliptic.   
      
   Albert    
      
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